## Abstract

In this note we consider a general compound approximation operator using binomial sequences and we give a representation for its corresponding remainder term. We also introduce a more general compound approximation operator using Sheffer sequences. We provide convergence theorems for both studied operators

## Authors

Maria **Craciun**

(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

## Keywords

Sequences of binomial type; Sheffer sequences, compound operators

## References

## About this paper

##### Cite this paper as:

M. Crăciun, *Compound operators constructed with binomial and Sheffer sequences*, Rev. Anal. Numér. Théor. Approx., vol. 32 (2003), no. 2, 135-144

##### Publisher Name

##### Paper on the journal website

##### Print ISSN

1222-9024

##### Online ISSN

2457-8126

##### Google Scholar Profile

[1] Agratini, O., *On a certain class of approximation operators*, Pure Math. Appl., 11, no. 2, pp. 119–127, 2001.

[2] Agratini, O., *Binomial polynomials and their applications in approximation theory*, Conf. Semin. Mat. Univ. Bari, 281, pp. 1–22, 2001.

[3] Craciun, M., *Approximation operators constructed by means of Sheffer sequences*, Rev. Anal. Numer. Theor. Approx., 30, no. 2, pp. 135–150, 2001.

[4] Craciun, M., *On an approximation operator and its Lipschitz constant*, Rev. Anal. Numer. Theor. Approx., 31, no. 1, pp. 55–60, 2002.

[5] Di Bucchianico, A., *Probabilistic and analytical aspects of the umbral calculus*, CWI Tract, 119, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1997.

[6] Di Bucchianico, A. and Loeb, D., *A selected survey of umbral calculus*, Electron. J. Combin., 2, Dynamic Survey 3, 28 pp. (electronic), 1995.

[7] Hildebrand, F. B., *Introduction to Numerical Analysis*, Second ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York Dusseldorf– Johannesburg, 1974.

[8] Lupas, A., *Approximation operators of binomial type*, Proc. IDoMAT 98, New developments in approximation theory, International Series of Numerical Mathematics, 132, Birkhauser Verlag, pp. 175–198, 1999.

[9] Lupas, L. and Lupas, A., *Polynomials of binomial type and approximation operators*, Studia Univ. Babes-Bolyai, Mathematica, 32, no. 4, pp. 40–44, 1987.

[10] Manole, C., *Expansions in series of generalized Appell polynomials with applications the approximation of functions*, PhD Thesis, “Babes-Bolyai” University, Cluj Napoca, 1984 (in Romanian)

[11] Manole, C., *Approximation operators of binomial type*, Research Seminar on Numerical and Statistical Calculus, Preprint no. 9, pp. 93–98, 1987.

[12] Mihesan, V., *Lipschitz constants for operators of binomial type of a Lipschitz continuous function*, RoGer 2000—Bra¸sov, pp. 81–87, Schr.reihe Fachbereichs Math. Gerhard Mercator Univ., 485, Gerhard–Mercator-Univ., Duisburg, 2000.

[13] Moldovan, G., *Sur la convergence de certains operateurs convolutifs positifs*, C. R. Acad. Sci. Paris Ser. A, 272, pp. 1311–1313, 1971 (in French).

[14] Popoviciu, T., *Remarques sur les polynomes binomiaux*, Bul. Soc. Stiinte Cluj, 6, pp. 146–148, 1931.

[15] Rota, G.-C., with the collaboration of P. Doubilet, C. Greene, D. Kahaner, A. Odlyzko and R. Stanley, *Finite Operator Calculus*, Academic Press, New York–London, 1975.

[16] Sablonniere, P., *Positive Bernstein–Sheffer operators*, J. Approx. Theory, 83, pp. 330–341, 1995.

[17] Shisha, O. and Mond, B., *The degree of convergence of linear positive operators*, Proc. Nat. Acad. Sci. USA, 60, pp. 1196–2000, 1968.

[18] Stancu, D. D., *Approximation of functions by means of a new generalized Bernstein operator*, Calcolo, 20, no. 2, pp. 211–229, 1983.

[19] Stancu, D. D., *A note on a multiparameter Bernstein-type approximating operator*, Mathematica (Cluj), 26 (49), no. 2, pp. 153–157, 1984.

[20] Stancu, D. D., *A note on the remainder in a polynomial approximation formula*, Studia Univ. Babes-Bolyai Math., 41, no. 2, pp. 95–101, 1996.

[21] Stancu, D. D., *Representation of remainders in approximation formulae by some discrete type linear positive operators*, Proceedings of the Third International Conference on Functional Analysis and Approximation Theory, Vol. II (Acquafredda di Maratea, 1996), Rend. Circ. Mat. Palermo (2) Suppl. No. 52, Vol. II, pp. 781–791, 1998.

[22] Stancu, D. D., *On approximation of functions by means of compound poweroid operators*, Mathematical Analysis and Approximation Theory, Proceedings of RoGer 2002- Sibiu, pp. 259–272.

[23] Stancu, D. D. and Drane, J. W., *Approximation of functions by means of the poweroid operators S α m,r,s*, Trends in approximation theory (Nashville, TN, 2000), pp. 401–405, Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2001.

[24] Stancu, D. D. and Occorsio, M. R., *On approximation by binomial operators of Tiberiu Popoviciu type*, Rev. Anal. Numer. Theor. Approx., 27, no. 1, pp. 167–181, 1998.

[25] Stancu, D. D. and Simoncelli, A. C., *Compound poweroid operators of approximation*, Proceedings of the Fourth International Conference on Functional Analysis and Approximation Theory (Acquafredda di Maratea, 2000), Rend. Circ. Mat. Palermo (2) Suppl., 68, pp. 845–854, 2002.

[26] Stancu, D. D. and Vernescu, A., *Approximation of bivariate functions by means of a class of operators of Tiberiu Popoviciu type*, Mathematical Reports, Bucure¸sti, (1) 51, no. 3, pp. 411–419, 1999.

[27] Steffensen, J. F., *Interpolation*, Chelsea, New York, 1927.

[28] Steffensen, J. F., *The poweroid, an extension of the mathematical notion of power*, Acta Math., 73, pp. 333–366, 1941.